Dirichlet series
L(s) = 1 | + 6.97e4·3-s − 5.64e9·7-s − 2.43e10·9-s − 2.87e12·13-s + 3.14e14·19-s − 3.93e14·21-s + 6.40e15·25-s − 5.73e15·27-s + 1.10e17·31-s − 7.52e17·37-s − 2.00e17·39-s + 1.30e18·43-s + 6.31e18·49-s + 2.19e19·57-s − 1.24e20·61-s + 1.37e20·63-s + 3.49e20·67-s − 6.55e19·73-s + 4.46e20·75-s + 3.25e20·79-s − 6.73e20·81-s + 1.62e22·91-s + 7.68e21·93-s − 2.39e22·97-s + 6.84e22·103-s − 6.47e22·109-s − 5.24e22·111-s + ⋯ |
L(s) = 1 | + 0.393·3-s − 2.85·7-s − 0.776·9-s − 1.60·13-s + 2.70·19-s − 1.12·21-s + 2.68·25-s − 1.03·27-s + 4.34·31-s − 4.22·37-s − 0.631·39-s + 1.40·43-s + 1.61·49-s + 1.06·57-s − 2.85·61-s + 2.21·63-s + 2.86·67-s − 0.209·73-s + 1.05·75-s + 0.434·79-s − 0.683·81-s + 4.58·91-s + 1.70·93-s − 3.34·97-s + 4.94·103-s − 2.50·109-s − 1.66·111-s + ⋯ |
Functional equation
Invariants
Degree: | \(16\) |
Conductor: | \(2^{32} \cdot 3^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(2.20660\times 10^{17}\) |
Root analytic conductor: | \(12.1334\) |
Motivic weight: | \(22\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [11]^{8} ),\ 1 )\) |
Particular Values
\(L(\frac{23}{2})\) | \(\approx\) | \(0.005596920630\) |
\(L(\frac12)\) | \(\approx\) | \(0.005596920630\) |
\(L(12)\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 2 | \( 1 \) |
3 | \( 1 - 23240 p T + 360822268 p^{4} T^{2} + 33772184600 p^{10} T^{3} + 242900943238 p^{20} T^{4} + 33772184600 p^{32} T^{5} + 360822268 p^{48} T^{6} - 23240 p^{67} T^{7} + p^{88} T^{8} \) | |
good | 5 | \( 1 - 1280906811524008 p T^{2} + \)\(90\!\cdots\!16\)\( p^{2} T^{4} - \)\(94\!\cdots\!56\)\( p^{7} T^{6} + \)\(83\!\cdots\!54\)\( p^{12} T^{8} - \)\(94\!\cdots\!56\)\( p^{51} T^{10} + \)\(90\!\cdots\!16\)\( p^{90} T^{12} - 1280906811524008 p^{133} T^{14} + p^{176} T^{16} \) |
7 | \( ( 1 + 2822790920 T + 1256624077538293444 p T^{2} + \)\(86\!\cdots\!00\)\( p^{4} T^{3} + \)\(22\!\cdots\!58\)\( p^{4} T^{4} + \)\(86\!\cdots\!00\)\( p^{26} T^{5} + 1256624077538293444 p^{45} T^{6} + 2822790920 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
11 | \( 1 - \)\(17\!\cdots\!80\)\( p T^{2} + \)\(58\!\cdots\!84\)\( p^{2} T^{4} + \)\(19\!\cdots\!60\)\( p^{3} T^{6} - \)\(14\!\cdots\!54\)\( p^{4} T^{8} + \)\(19\!\cdots\!60\)\( p^{47} T^{10} + \)\(58\!\cdots\!84\)\( p^{90} T^{12} - \)\(17\!\cdots\!80\)\( p^{133} T^{14} + p^{176} T^{16} \) | |
13 | \( ( 1 + 1437841190680 T + \)\(68\!\cdots\!76\)\( T^{2} + \)\(11\!\cdots\!20\)\( p T^{3} + \)\(14\!\cdots\!14\)\( p^{2} T^{4} + \)\(11\!\cdots\!20\)\( p^{23} T^{5} + \)\(68\!\cdots\!76\)\( p^{44} T^{6} + 1437841190680 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
17 | \( 1 - \)\(68\!\cdots\!20\)\( T^{2} + \)\(77\!\cdots\!36\)\( p^{2} T^{4} - \)\(55\!\cdots\!60\)\( p^{4} T^{6} + \)\(27\!\cdots\!94\)\( p^{6} T^{8} - \)\(55\!\cdots\!60\)\( p^{48} T^{10} + \)\(77\!\cdots\!36\)\( p^{90} T^{12} - \)\(68\!\cdots\!20\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
19 | \( ( 1 - 157401978775768 T + \)\(12\!\cdots\!92\)\( p T^{2} - \)\(42\!\cdots\!16\)\( p^{2} T^{3} + \)\(20\!\cdots\!10\)\( p^{3} T^{4} - \)\(42\!\cdots\!16\)\( p^{24} T^{5} + \)\(12\!\cdots\!92\)\( p^{45} T^{6} - 157401978775768 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
23 | \( 1 - \)\(26\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!84\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{8} - \)\(31\!\cdots\!80\)\( p^{44} T^{10} + \)\(29\!\cdots\!84\)\( p^{88} T^{12} - \)\(26\!\cdots\!60\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
29 | \( 1 - \)\(10\!\cdots\!08\)\( T^{2} + \)\(47\!\cdots\!48\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!76\)\( p^{44} T^{10} + \)\(47\!\cdots\!48\)\( p^{88} T^{12} - \)\(10\!\cdots\!08\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
31 | \( ( 1 - 55146614124607288 T + \)\(28\!\cdots\!48\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!96\)\( p^{22} T^{5} + \)\(28\!\cdots\!48\)\( p^{44} T^{6} - 55146614124607288 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
37 | \( ( 1 + 376245183281642200 T + \)\(13\!\cdots\!88\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!98\)\( T^{4} + \)\(27\!\cdots\!40\)\( p^{22} T^{5} + \)\(13\!\cdots\!88\)\( p^{44} T^{6} + 376245183281642200 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
41 | \( 1 - \)\(15\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!24\)\( T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!86\)\( T^{8} - \)\(59\!\cdots\!20\)\( p^{44} T^{10} + \)\(11\!\cdots\!24\)\( p^{88} T^{12} - \)\(15\!\cdots\!40\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
43 | \( ( 1 - 650507781721037080 T + \)\(13\!\cdots\!04\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} - \)\(10\!\cdots\!60\)\( p^{22} T^{5} + \)\(13\!\cdots\!04\)\( p^{44} T^{6} - 650507781721037080 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
47 | \( 1 - \)\(20\!\cdots\!72\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{4} - \)\(14\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!24\)\( p^{44} T^{10} + \)\(19\!\cdots\!68\)\( p^{88} T^{12} - \)\(20\!\cdots\!72\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
53 | \( 1 - \)\(40\!\cdots\!40\)\( T^{2} + \)\(84\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!20\)\( p^{44} T^{10} + \)\(84\!\cdots\!44\)\( p^{88} T^{12} - \)\(40\!\cdots\!40\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
59 | \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(74\!\cdots\!64\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!80\)\( p^{44} T^{10} + \)\(74\!\cdots\!64\)\( p^{88} T^{12} - \)\(35\!\cdots\!60\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
61 | \( ( 1 + 62207640705914143384 T + \)\(70\!\cdots\!60\)\( T^{2} + \)\(31\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!74\)\( T^{4} + \)\(31\!\cdots\!96\)\( p^{22} T^{5} + \)\(70\!\cdots\!60\)\( p^{44} T^{6} + 62207640705914143384 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
67 | \( ( 1 - \)\(17\!\cdots\!40\)\( T + \)\(39\!\cdots\!08\)\( T^{2} - \)\(32\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!18\)\( T^{4} - \)\(32\!\cdots\!80\)\( p^{22} T^{5} + \)\(39\!\cdots\!08\)\( p^{44} T^{6} - \)\(17\!\cdots\!40\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
71 | \( 1 - \)\(93\!\cdots\!08\)\( T^{2} + \)\(13\!\cdots\!48\)\( T^{4} - \)\(79\!\cdots\!76\)\( T^{6} + \)\(60\!\cdots\!70\)\( T^{8} - \)\(79\!\cdots\!76\)\( p^{44} T^{10} + \)\(13\!\cdots\!48\)\( p^{88} T^{12} - \)\(93\!\cdots\!08\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
73 | \( ( 1 + 32788189594240031800 T + \)\(37\!\cdots\!96\)\( T^{2} + \)\(91\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!86\)\( T^{4} + \)\(91\!\cdots\!00\)\( p^{22} T^{5} + \)\(37\!\cdots\!96\)\( p^{44} T^{6} + 32788189594240031800 p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
79 | \( ( 1 - \)\(16\!\cdots\!72\)\( T + \)\(16\!\cdots\!28\)\( T^{2} - \)\(26\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} - \)\(26\!\cdots\!04\)\( p^{22} T^{5} + \)\(16\!\cdots\!28\)\( p^{44} T^{6} - \)\(16\!\cdots\!72\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
83 | \( 1 - \)\(98\!\cdots\!24\)\( T^{2} + \)\(44\!\cdots\!80\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{6} + \)\(24\!\cdots\!54\)\( T^{8} - \)\(12\!\cdots\!36\)\( p^{44} T^{10} + \)\(44\!\cdots\!80\)\( p^{88} T^{12} - \)\(98\!\cdots\!24\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
89 | \( 1 - \)\(31\!\cdots\!80\)\( T^{2} + \)\(65\!\cdots\!56\)\( p T^{4} - \)\(68\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!26\)\( T^{8} - \)\(68\!\cdots\!40\)\( p^{44} T^{10} + \)\(65\!\cdots\!56\)\( p^{89} T^{12} - \)\(31\!\cdots\!80\)\( p^{132} T^{14} + p^{176} T^{16} \) | |
97 | \( ( 1 + \)\(11\!\cdots\!60\)\( T + \)\(25\!\cdots\!24\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!66\)\( T^{4} + \)\(19\!\cdots\!80\)\( p^{22} T^{5} + \)\(25\!\cdots\!24\)\( p^{44} T^{6} + \)\(11\!\cdots\!60\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \) | |
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Imaginary part of the first few zeros on the critical line
−3.77703419788416203260738809902, −3.48665970565666092866602438177, −3.18886863656752654582117664681, −3.16780436366385532219631100748, −3.15385481680902465148332484438, −3.14260913464578516409114135010, −3.07981612240050313979833946135, −2.90283601669805379241204317345, −2.87159023593501464058333233768, −2.43460270159598717344999419416, −2.18933111022870145581059649614, −2.17890029143368922044907673271, −1.96916007705604828660007024405, −1.93844931486635916362688001003, −1.92749725708643365104340225708, −1.20333853749987634761540058450, −1.15510460053233031884093582302, −1.14274924159614767290185487336, −0.976961342663307278810987760999, −0.967038777231574121194043039339, −0.812464410544177050466230045789, −0.48967993596125555018132986279, −0.33108627430124348490269639967, −0.04283234793107812496942579412, −0.02869567705358515579870533413, 0.02869567705358515579870533413, 0.04283234793107812496942579412, 0.33108627430124348490269639967, 0.48967993596125555018132986279, 0.812464410544177050466230045789, 0.967038777231574121194043039339, 0.976961342663307278810987760999, 1.14274924159614767290185487336, 1.15510460053233031884093582302, 1.20333853749987634761540058450, 1.92749725708643365104340225708, 1.93844931486635916362688001003, 1.96916007705604828660007024405, 2.17890029143368922044907673271, 2.18933111022870145581059649614, 2.43460270159598717344999419416, 2.87159023593501464058333233768, 2.90283601669805379241204317345, 3.07981612240050313979833946135, 3.14260913464578516409114135010, 3.15385481680902465148332484438, 3.16780436366385532219631100748, 3.18886863656752654582117664681, 3.48665970565666092866602438177, 3.77703419788416203260738809902